Question

A 1024-bit message is sent that contains 992 data bits and 32 CRC bits. CRC is computed using the IEEE 802 standardized, 32-degree CRC polynomial. For each of the following, explain whether the errors during message transmission will be detected by the receiver: (a) There was a single-bit error. (b) There were two isolated bit errors. (c) There were 18 isolated bit errors. (d) There were 47 isolated bit errors. (e) There was a 24-bit long burst error. (f) There was a 35 -bit long burst error.

Short answer

CRC-32 will detect errors in all cases except possibly (f), where a 35-bit burst error is beyond the guaranteed detection capability.

Step-by-step solution

Understanding CRC Detection Capabilities

The CRC-32 polynomial, used in IEEE 802 standards, can perfectly detect any single-bit errors, any double-bit errors, any odd number of errors, and burst errors up to 32 bits in length. It has a high probability of detecting other errors, but these are the guaranteed detections.

Evaluating Single-Bit Error (a)

A single-bit error changes only one bit of the data. The CRC-32 can detect all single-bit errors because it automatically changes the remainder produced by the polynomial division in the CRC algorithm.

Evaluating Two Isolated Bit Errors (b)

Two isolated bit errors mean there are two separate bit flips in the message. CRC-32 can detect all double isolated bit errors as they cause the remainder to differ from the expected result.

Evaluating 18 Isolated Bit Errors (c)

Eighteen isolated bit errors result in an odd number of errors in the message. CRC-32, which can detect all odd numbers of bit errors, will identify this error.

Evaluating 47 Isolated Bit Errors (d)

Forty-seven isolated bit errors is also an odd number of errors, which CRC-32 can detect. Error detection is guaranteed in this case as well.

Evaluating 24-bit Long Burst Error (e)

A 24-bit long burst error is well under the 32-bit limit that CRC-32 can reliably detect for burst errors. This length of burst error will be detected.

Evaluating 35-bit Long Burst Error (f)

A 35-bit long burst error exceeds the maximum guaranteed detectable length for burst errors with CRC-32. While there is still a high probability it will be detected, detection is not guaranteed.

Key concepts

CRC-32 Polynomial

CRC-32, or Cyclic Redundancy Check 32, is an error-detecting code used to ensure data integrity. It works by generating a short, fixed-length binary sequence based on the input data. This sequence is called a checksum. CRC-32 utilizes a specific mathematical algorithm, the CRC-32 polynomial, which is a polynomial of degree 32. This polynomial is standardized in the IEEE 802 standards and is used across many networking protocols to help maintain accurate data transmission.
The key idea behind CRC-32 is to perform binary division of the message by this polynomial. The remainder of this division acts as a fingerprint for the data. If any bits in the message are changed, the remainder changes, indicating an error. CRC-32 is particularly effective because it can detect changes up to a certain length of errors like burst errors, and it is well-suited for detecting random errors.

Error Detection

Error detection is a fundamental aspect of digital communication. It ensures that data received is identical to data sent. CRC-32 is a robust tool in the realm of error detection due to its capability to detect various types of errors.
Here’s what CRC-32 can detect:

• All single-bit errors, since they change the binary division remainder.
• All double-bit errors, making it reliable for catching simple mistakes.
• Any odd number of errors, which alters the checksum and is detectable.
• Burst errors up to 32 bits long, which CRC-32 can guarantee to catch.

While CRC-32 is powerful, it doesn't provide 100% certainty for very long or complex error patterns, but it drastically reduces the likelihood of undetected errors.

Burst Errors

Burst errors occur when multiple bits in a sequence are altered simultaneously. These errors are common in network data transmissions when there are sudden noise spikes.
CRC-32 excels at detecting these types of errors, especially important in maintaining the fidelity of network communications. A notable feature of CRC-32 is its ability to perfectly detect burst errors up to 32 bits in length. This means if a burst error involves altering bits consecutively, CRC-32 can pick it up as long as it's within this limit. For errors longer than 32 bits, CRC can still detect them, but the guarantee isn’t as strong.
Therefore, CRC-32 provides a significant level of reassurance in the presence of such transmission disturbances.

IEEE 802 Standards

The IEEE 802 standards play a critical role in networking, dictating how data is formatted and transmitted across various media. CRC-32 is an integral part of these standards, especially within Ethernet and Wi-Fi communications, ensuring data packets remain unchanged from source to destination. These standards are crucial for maintaining compatibility and reliability across different devices and technologies.
Within these standards, CRC-32 is frequently employed to append a checksum to data packets. This method aids in error detection by confirming that data has been accurately received. If the recalculated checksum at the destination doesn't match the one transmitted, an error is flagged. This mechanism supports seamless communication across global networks and fortifies the robustness of internet communications.